hw5_sol.pdf - CO 370 Homework assignment 5 Fall 17 Page 1...

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CO 370 - Homework assignment 5 Fall ’17 Page 1 CO 370 - Fall ’17 Homework assignment #5: (Due on Friday, Dec 1, at 11:59pm) Instructions: You will be graded not only on correctness, but in clarity of exposition. Allowed sources of help: CO370 course notes and your CO370 classmates. Please write down the name of the people you collaborate with. Any source of help other than the ones stated above are considered cheating. In the models, please define clearly what are your decision variables. Particularly, clearly distinguish first and second-stage decision variables. Also, state clearly what your con- straints represent. All answers must be justified unless otherwise stated. You are responsible to make sure your submission on crowdmark is legible by the grader and that it is received on time. Submissions that are poorly scanned may be penalized depending on how bad the scan is. Question 1 (30 points) John must commit, at the beginning of the day, how much of each products 1,2,3,4 to sell. The per unit profit of selling products 1, 2, 3, 4 are, respectively, $2, $1, $3, $5. Producing one unit of each product requires a certain amount of a given resource, which John has only 25 units available. However, the amount of the resource that is consumed by producing product j is uncertain (for each j = 1 , . . . , 4). (a) (10 points) Consider that the amount of the resource required for producing one unit of product j is ˜ a j . Consider now that the uncertainty set of possible coefficients is: PA = ˜ a R 4 + : a 1 + 4˜ a 2 + ˜ a 3 + ˜ a 4 16 Write down an LP that can be used to solve the problem that John has of finding how much of each product to produce so that, for any possible constraint coefficients ˜ a John will never exceed the 25 units he has available. Show your work. Solution: The problem we are interested in can be written as: max 2 x 1 + x 2 + 3 x 3 + 5 x 4 s.t. 4 j =1 ˜ a j x j 25 , ˜ a PA x 0 (ROB) where decision variable x j represents how much of product j that John commits to selling. The nontrivial constraints in (ROB) can be equivalently written as (for a fixed ¯ x ): max 4 j =1 ˜ a j ¯ x j s.t. a 1 + 4˜ a 2 + ˜ a 3 + ˜ a 4 16 ˜ a 0 25 (1)
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CO 370 - Homework assignment 5 Fall ’17 Page 2 (1) is always feasible and bounded, thus it has an optimal solution, thus by strong duality, we can rewrite (1) as min 16 α s.t 2 α ¯ x 1 4 α ¯ x 2 α ¯ x 3 α ¯ x 4 α 0 25 (2) By a result seen in class, we can thus rewrite (ROB) as the following LP max 2 x 1 + x 2 + 3 x 3 + 5 x 4 s.t. 16 α 25 2 α x 1 4 α x 2 α x 3 α x 4 α 0 x 0 (ROB-LP) (b) (10 points) Consider now that there are five possible scenarios for the set of coefficients ˜ a , each can happen with a probability of 0.2: 1. ˜ a = (1 , 1 , 1 , 9) 2. ˜ a = (1 , 1 , 9 , 1) 3. ˜ a = (5 , 1 , 1 , 1) 4. ˜ a = (1 , 3 , 1 , 1) 5. ˜ a = (1 , 1 , 1 , 1) John decides to write a two-stage stochastic program to decide how much to produce per day (in advance, before knowing ˜ a ). If the decision that John takes exceeds the total available units of the
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  • Winter '11
  • henry
  • Optimization, Approximation algorithm, optimization problem

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